arXiv research

Paper shows how to evenly distribute intersections in hyperbolic spaces.

In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…

Study properties of self-similar continua with finite intersection property.

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…

The paper proves the exact number of singular points in the intersection of convex shapes.

The paper provides an algorithm to create curves touching a smooth cubic at specific intersection points.

Computes expected number of real intersection points of essential variety with random linear spaces.

Suppose a smooth planar curve $γ$ is $2π$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $γ$ self-intersects, then it has a segment of length $\ell- 2π$ on which it self-intersects and somewhere its curvature is at least $2π/(\ell - 2π)$. The proof involves the projection $Γ$ …

Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.

Method detects intersections between ellipses for Borromean linking.

A new method clusters intersecting lines using hypergraphs.

The paper studies the number of normals to ellipsoids and their intersections with caustics.

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…

Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.

We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect the Seifert surface.

Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of special lagrangian and lagrangian varieties in certain homogeneous spaces.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.

A new proof shows almost every normal to a smooth convex body intersects at least 6 normals from different points.

Geodesics on hyperbolic surfaces become evenly spread over time.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

Novel approach for large genus intersection number asymptotics.

In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G= F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of $G$.

The paper proves that any smooth curve can have two similar inscribed rectangles.

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the cla…

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems r…

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…

Study on symmetry defects of complete intersections in complex space.

Harmonic maps intersect all minimal surfaces with bounded curvature.

Probabilistic theory counts intersections in Riemannian spaces.

We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being …

Formula counts rational curves with a specific singular point in projective space.

A filling curve $γ$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(γ)$ of $S$ via the process of "point-pushing along $γ$." We consider the relationship between the self-intersection number $i(γ)$ of $γ$ and the dilatation of $P(γ)$; our main result is that the dilatation is bounded between $(i(γ)+1…

The study finds that most minimal surfaces in generic 4D manifolds intersect in complex ways.

The study proves geodesic loops and chords without intersections for specific metrics.

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

Constructs geodesics near intersection points of Lagrangian submanifolds.

Study minimizes crossing points of up to 12 curves on a genus 2 surface.

Unified model for knot polynomials using quantum Heegaard diagrams.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersecti…

The paper classifies vertices in planar polygons formed by convex domains.

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by consi…

The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…

This paper offers a new algebraic perspective of GCCA using subspace intersection.